How to understand this decomposition of the spectrum of a QFT The reference for this question is the following set of notes on the conformal bootstrap, section 1.2.2. Suppose we have a quantum field theory with symmetry algebra $\mathfrak g$ and spectrum $\mathcal S$. It is then stated that we can decompose the spectrum as follows $$\mathcal S = \bigoplus_{\mathcal R} m_{\mathcal R}\mathcal R,$$ where $\mathcal R$ are the representations of this symmetry algebra and $m_{\mathcal R}\in \mathbb N\cup \{\infty\}$   is the multiplicity of the representation $\mathcal R$. However, these two things on either side of the equal sign are two completely different things. On the LHS, we have a vector space, and on the RHS we have a direct sum over all such representations of the symmetry algebra of this vector space (i.e., all homomorphisms $\mathcal R:\mathfrak g\to \mathfrak{gl}(V)$, for $V$ a vector space). How can one then make sense of this equality, seeing as these objects are so different?
 A: Let $G$ be a group. Most of the time in Physics will be a Lie group but it need not be for this discussion. A representation $G$ is a pair $(V,\rho)$ where $V$ is a vector space and $\rho:G\to{\rm GL}(V)$ is a group homomorphism. It maps elements of $G$ into invertible linear operators acting on $V$.
In particular, we can define the direct sum of representations. Let $(V,\rho)$ and $(V',\rho')$ be two representations. The direct sum is the representation $(V\oplus V',\rho\oplus \rho')$.
$$(\rho\oplus \rho')(g) (v,w)=(\rho(g)v,\rho'(g)w).\tag{1}$$
The same construction can be carried out for Lie algebras $\mathfrak{g}$ and their representations.
The point is that both sides of your equality are representations, and we have a definition for the direct sum of two representations: we take the direct sum of the vector spaces and endow it with the group action defined in (1) or its Lie algebra analog in the case of representation of algebras.
In particular, your equality means that the vector space on the right-hand side decomposes as a direct sum of the vector spaces of the representations appearing on the right-hand side.
